When The Rifleman's Journal was still up, you could find some primer testing (the Wayback machine still has it). With LR, it didn't generally make a dramatic velocity or pressure difference. With SR there is more difference in some tests, up to the equivalent of about 5% of the powder charge in one test Charles Petty did in Handloader in 2006. That was with a 55-grain V-max in a 223 using Reloader 10X. It ran 3150 fps from a bolt rifle using Federal 205 primers, which was the mildest primer he tried, and 3300 fps using the hottest primer he tried (I've forgotten which one that was, as it was awhile back). Anyway, in QuickLOAD that a velocity difference is also generated with about a 5% increase in powder charge in a 22" barrel.
Another situation that can arise is SD can be lower with a standard primer when the case is pretty well filled with powder but then be lower with a magnum primer when the loading density is low. The magnum primers often do better with Garand loads, which are notoriously low in loading density. But it depends on the powder choice, too. There isn't a fixed rule about it.
